Real-time well trajectory projection using stochastic processes

ABSTRACT

Systems and methods for stochastically projecting a well trajectory of a bottom hole assembly in a subsurface formation, where the bottom hole assembly includes one or more transducers, a trajectory controller coupled to the bottom hole assembly, an information handling system coupled to the transducers, and the information system includes a processor, and a non-transitory computer readable medium for storing one or more instructions that, when executed, causes the processor to receive a first one or more system model parameters from a system model parameter probability distribution; receive a first one or more steering inputs; receive a first one or more values corresponding to the bottom hole assembly initial conditions from the one or more transducers at a first position within a subsurface formation; and stochastically project a trajectory of the bottom hole assembly from the first position within the subsurface formation to a second position within the subsurface formation.

FIELD OF THE INVENTION

The present disclosure relates to a system and methods for projectingthe trajectory of a drilling assembly in a subsurface formation, andmore specifically to systems and methods for generating stochastictrajectory projections in real-time to predict the movement of a bottomhole assembly coupled to a drill string across a depth horizon andthereby improve control of the bottom hole assembly across the depthhorizon.

BACKGROUND

Boreholes drilled into subsurface formations may enable recovery ofdesirable fluids, including, without limitation, hydrocarbons, using anynumber of different techniques. In drilling operations, typical drillingprocesses may be relatively complex and involve considerable expense.Many of these drilling operations may be done manually with experiencedoperators running the drilling platform. There are continual efforts toimprove safety, improve fluid recovery, and lower costs associated withsubsurface drilling and advancements in computerized and automatedsystems in drilling processes may support these efforts.

Model-based control methods are now widely utilized to control thetrajectory of borehole placement during exploration of and extractionoperations in subsurface formations. Due to the complexity anduncertainty in drilling operations, it is challenging to find effectivemodels for control. High-fidelity models have been established in thepast, but often cannot be used for real-time dynamic control ofsubsurface drilling operations as these high-fidelity models aregenerally high dimension and computationally expensive, thus cannot beused in real-time. Reduced physics-based models have also beendeveloped. These reduced physics-based models are simpler and mayprovide more confidence for a short range that may be suitable forreal-time control if they are updated frequently using the measurementsfrom subsurface equipment. However, due to uncertainties in the bit-rockinteractions, drilling parameter changes, sensor noise or malfunctions,downhole vibrations, and model/system discrepancies, reducedphysics-based models with deterministic parameters may not be sufficientfor real-time control of drilling operations.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the present embodiments and advantagesthereof may be acquired by referring to the following description takenin conjunction with the accompanying drawings, in which like referencenumbers indicate like features.

FIG. 1 depicts an exemplary drilling system.

FIG. 2 depicts an exemplary flow diagram illustrating a method fordetermining one or more stochastic trajectory projections and confidenceregions for a bottom hole assembly based at least in part on one or moreof system model parameters, steering inputs, bottom hole assemblyinitial conditions, working mode selection and settings, and a desirednumber of trajectory projections.

FIG. 3 depicts an exemplary well plan and stochastic trajectoryconfidence regions for a borehole across a depth horizon.

FIGS. 4-5 depict exemplary stochastic trajectory projections andconfidence regions for a bottom hole assembly across a depth horizon.FIGS. 6 a-b depicts two exemplary stochastic trajectory projections andconfidence regions for a bottom hole assembly across a depth horizon.

FIG. 7 depicts an exemplary flow diagram for generating stochastictrajectory projections for a bottom hole assembly.

FIG. 8 depicts a schematic diagram of an information handling system foruse with or in a wellbore environment, according to one or more aspectsof the present disclosure.

While embodiments of this disclosure have been depicted and describedand are defined by reference to exemplary embodiments of the disclosure,such references do not imply a limitation on the disclosure and no suchlimitation is to be inferred. The subject matter disclosed is capable ofconsiderable modification, alteration, and equivalents in form andfunction, as will occur to those skilled in the pertinent art and havingthe benefit of this disclosure. The depicted and described embodimentsof this disclosure are examples only and are not exhaustive of the scopeof the disclosure.

DETAILED DESCRIPTION

The present disclosure relates to a system and methods for projectingthe trajectory of a drilling assembly in a subsurface formation, andmore specifically to systems and methods for generating stochastictrajectory projections in real-time to predict the movement of a bottomhole assembly coupled to a drill string across a depth horizon andthereby improve control of the bottom hole assembly across the depthhorizon.

The system and method disclosed herein uses a stochastic trajectoryprojection module. The stochastic trajectory projection module may use aplurality of inputs to generate a projected trajectory and confidenceregions for a bottom hole assembly across a depth horizon in asubsurface formation. The stochastic trajectory projection module mayuse one or more stochastic models, including, without limitation, MonteCarlo simulation methods, to simulate and project the futuretrajectories. The stochastic trajectory projection module disclosedherein supports two modes, where selection of the mode and anycorresponding settings may be dependent on any prior data analyses,including without limitation higher fidelity models, or knowledge of oneor more of the subsurface formation and the equipment of the drillingsystem.

The stochastic trajectory projection module further enables real-timeprobabilistic projections for trajectories and corresponding confidenceregions based on system model parameters, steering inputs, and bottomhole assembly initial conditions.

The system and method disclosed herein provide a unique way to projectthe borehole trajectories. This enables drilling personnel and steeringcontrol systems to plan ahead and improve steering decisions, resultingin improved well placement and, thereby, improving fluid recovery andlowering costs associated with subsurface drilling operations. Asdiscussed herein, real-time trajectory projections enable feedbackduring drilling operations that enables an operator to refine thesteering inputs to the drilling operation equipment duringmeasurement-while drilling (MWD) or logging-while-drilling (LWD)operations.

In one or more aspects of the present disclosure, a borehole environmentmay utilize an information handling system to control one or moreoperations associated with the borehole environment. For purposes ofthis disclosure, an information handling system may include anyinstrumentality or aggregate of instrumentalities operable to compute,classify, process, transmit, receive, retrieve, originate, switch,store, display, manifest, detect, record, reproduce, handle, or utilizeany form of information, intelligence, or data for business, scientific,control, or other purposes. For example, an information handling systemmay be a personal computer, a network storage device, or any othersuitable device and may vary in size, shape, performance, functionality,and price. The information handling system may include random accessmemory (RAM), one or more processing resources such as a centralprocessing unit (CPU) or hardware or software control logic, ROM, and/orother types of nonvolatile memory. Additional components of theinformation handling system may include one or more disk drives, one ormore network ports for communication with external devices as well asvarious input and output (I/O) devices, such as a keyboard, a mouse, anda video display. The information handling system may also include one ormore buses operable to transmit communications between the varioushardware components. The information handling system may also includeone or more interface units capable of transmitting one or more signalsto a controller, actuator, or like device.

For the purposes of this disclosure, computer-readable media may includeany instrumentality or aggregation of instrumentalities that may retaindata and/or instructions for a period of time. Computer-readable mediamay include, for example, without limitation, storage media such as asequential access storage device (for example, a tape drive), directaccess storage device (for example, a hard disk drive or floppy diskdrive), compact disk (CD), CD read-only memory (ROM) or CD-ROM, DVD,RAM, ROM, electrically erasable programmable read-only memory (EEPROM),and/or flash memory, biological memory, molecular or deoxyribonucleicacid (DNA) memory as well as communications media such wires, opticalfibers, microwaves, radio waves, and other electromagnetic and/oroptical carriers; and/or any combination of the foregoing.

Illustrative embodiments of the present disclosure are described indetail herein. In the interest of clarity, not all features of an actualimplementation may be described in this specification. It will of coursebe appreciated that in the development of any such actual embodiment,numerous implementation-specific decisions may be made to achieve thespecific implementation goals, which may vary from one implementation toanother. Moreover, it will be appreciated that such a development effortmight be complex and time-consuming, but would nevertheless be a routineundertaking for those of ordinary skill in the art having the benefit ofthe present disclosure.

Throughout this disclosure, a reference numeral followed by analphabetical character refers to a specific instance of an element andthe reference numeral alone refers to the element generically orcollectively. Thus, as an example (not shown in the drawings), widget “la” refers to an instance of a widget class, which may be referred tocollectively as widgets “1” and any one of which may be referred togenerically as a widget “1”. In the figures and the description, likenumerals are intended to represent like elements.

To facilitate a better understanding of the present disclosure, thefollowing examples of certain embodiments are given. In no way shouldthe following examples be read to limit, or define, the scope of thedisclosure. Embodiments of the present disclosure may be applicable todrilling operations that include but are not limited to target (such asan adjacent well) following, target intersecting, target locating, welltwinning such as in SAGD (steam assist gravity drainage) wellstructures, drilling relief wells for blowout wells, river crossings,construction tunneling, as well as horizontal, vertical, deviated,multilateral, u-tube connection, intersection, bypass (drill around amid-depth stuck fish and back into the well below), or otherwisenonlinear boreholes in any type of subsurface formation. Embodiments maybe applicable to injection wells, and production wells, includingnatural resource production wells such as hydrogen sulfide, hydrocarbonsor geothermal wells; as well as wellbore or borehole construction forriver crossing tunneling and other such tunneling boreholes for nearsurface construction purposes or borehole u-tube pipelines used for thetransportation of fluids such as hydrocarbons. Embodiments describedbelow with respect to one implementation are not intended to belimiting.

FIG. 1 depicts an exemplary drilling system 100. As depicted, borehole102 may extend from a wellhead 104 into a subsurface formation 106 froma surface 108. As depicted in FIG. 1 , in one or more embodiments,borehole 102 may extend generally vertically into the subsurfaceformation 106. Alternatively, in one or more embodiments, borehole 102may extend at an angle through subsurface formation 106, such ashorizontal and slanted boreholes. For example and without limitation,although FIG. 1 depicts a vertical or low inclination angle well, in oneor more embodiments, a high inclination angle or horizontal placement ofthe well and equipment may be possible. In one or more embodiments,borehole 102 may comprise any one or more of horizontal, vertical,slanted, curved, and any other types of borehole geometries andorientations. Borehole 102 may be cased or uncased. in one or moreembodiments, borehole 102 may include a metallic member, wherein themetallic member may be a casing, liner, tubing, or other elongated steeltubular disposed in borehole 102. While FIG. 1 generally depicts aland-based system, it should be noted that like systems may operate insubsea locations as well.

As depicted in FIG. 1 , a drilling platform 110 may support a derrick112 having a traveling block 114 for raising and lowering drill string116. Drill string 116 may include, but is not limited to, drill pipe andcoiled tubing, as generally known to those skilled in the art. A kelly118 may support drill string 116 as it may be lowered through a rotarytable 120. A drill bit 122 may be attached to the distal end of drillstring 116 and may be driven either by one or more of a downhole motorand rotation of drill string 116 from surface 108. For example andwithout limitation, drill bit 122 may include one or more of roller conebits, PDC bits, natural diamond bits, hole openers, reamers, coringbits, and the like. As drill bit 122 rotates, it may create and extendborehole 102 that penetrates various subsurface formations 106. A pump124 may circulate drilling fluid through a feed pipe 126 through kelly118, downhole through interior of drill string 116, through orifices indrill bit 122, back to surface 108 via annulus 128 surrounding drillstring 116, and into a retention pit 132.

With continued reference to FIG. 1 , drill string 116 may begin atwellhead 104 and may traverse borehole 102. Drill bit 122 may beattached to a distal end of drill string 116 and may be driven, orexample and without limitation, either by a downhole motor and/or viarotation of drill string 116 from surface 108. Drill bit 122 may be apart of bottom hole assembly 130 at the distal end of drill string 116.Bottom hole assembly 130 may further include tools for look-aheadresistivity applications. As will be appreciated by those of ordinaryskill in the art, bottom hole assembly 130 may be a measurement-whiledrilling or logging-while-drilling system.

Bottom hole assembly 130 may comprise any one or more of tools,transmitters, and receivers to perform downhole measurement operations.For example and without limitation, bottom hole assembly 130 maycomprise one or more of any number of assemblies for one or more ofmeasurement, communication, energy storage, and the like. For exampleand without limitation, bottom hole assembly 130 may comprisemeasurement assembly 134. In one or more embodiments, measurementassembly 134 may comprise at least one transducer 136 a, which may bedisposed at the surface of measurement assembly 134. While FIG. 1depicts a single transducer 136 a, in one or more embodiments, there maybe any number of transducers disposed on measurement assembly 134.References to and illustrations showing any one or more of transducers136 a-c may be applicable to any transducers disclosed herein. Withoutlimitation, transducers may be referred to herein as a transceiver andtransducer 136 a may be disposed within measurement assembly 134. In oneor more embodiments, measurement assembly 134 may further comprise fourother transducers that may be disposed ninety degrees from each other.In one or more embodiments, any number of transducers may be disposedalong bottom hole assembly 130 at any degree from each other. In one ormore embodiments, transducer 136 a, and any other transducer, mayfunction and operate to generate an acoustic pressure pulse that travelsthrough one or more borehole fluids. In one or more embodiments,transducers 136 a may further sense and acquire the reflected pressurewave which is modulated (for example and without limitation, reflectedas an echo by the borehole wall). During measurement operations, thetravel time of the pulse wave from transmission to recording of the echomay be recorded. In one or more embodiments, the acquired informationmay be used to determining for example and without limitation, a radiusof the borehole, which may be derived by the fluid sound speed. Byanalyzing the amplitude of the echo signal, the acoustic impedance mayalso be derived. In one or more embodiments, transducers 136 a may bemade of piezo-ceramic crystals, magnetostrictive materials, or any othermaterials that generate an acoustic pulse when activated, eitherelectrically or otherwise. In one or more embodiments, transducers 136 amay also include backing materials and matching layers. In one or moreembodiments, transducers 136 a and assemblies housing transducers 136 amay be removable and replaceable, for example and without limitation, inthe event of damage or failure.

In one or more embodiments, bottom hole assembly 130 may be one or moreof coupled to and controlled by information handling system 138, whichmay be disposed on surface 108. In one or more embodiments, informationhandling system 138 may be disposed down hole in bottom hole assembly130. Processing of information recorded may occur at one or more of downhole and on surface 108. Processing occurring downhole may betransmitted to surface 108 to be one or more of recorded, observed, andfurther analyzed. In one or more embodiments, information recorded oninformation handling system 138 that may be disposed down hole may bestored until bottom hole assembly 130 may be brought to surface 108. Inone or more embodiments, information handling system 138 may communicatewith bottom hole assembly 130 through a communication line (not shown)disposed in or on drill string 116. In one or more embodiments, wirelesscommunication may be used to transmit information back and forth betweeninformation handling system 138 and bottom hole assembly 130.Information handling system 138 may transmit information to bottom holeassembly 130 and may receive as well as process information recorded bybottom hole assembly 130. In one or more embodiments, a downholeinformation handling system (not shown) may include suitable circuitry,for example and without limitation, a microprocessor, for estimating,receiving, and processing signals from bottom hole assembly 130.Downhole information handling system (not shown) may further compriseone or more of additional components, including, without limitation,memory, input devices, output devices, interfaces, and the like. In oneor more embodiments, while not shown, bottom hole assembly 130 mayinclude one or more additional components, including, withoutlimitation, analog-to-digital converters, filters, and amplifiers, amongothers, that may be used to process the measurements of bottom holeassembly 130 before they may be transmitted to surface 108. In one ormore embodiments, raw measurements from bottom hole assembly 130 may betransmitted to surface 108.

Any suitable technique may be used for transmitting signals from bottomhole assembly 130 to surface 108, including, without limitation, wiredpipe telemetry, mud-pulse telemetry, acoustic telemetry, andelectromagnetic telemetry. While not shown, bottom hole assembly 130 mayinclude a telemetry subassembly that may transmit telemetry data tosurface 108. At surface 108, pressure transducers (not shown) mayconvert the pressure signal into electrical signals for a digitizer (notshown). The digitizer may supply a digital form of the telemetry signalsto information handling system 138 via a communication link 140, whichmay be a wired or wireless link. The telemetry data may be analyzed andprocessed by information handling system 138.

As depicted in FIG. 1 , communication link 140 (which may be wired orwireless, for example) may be provided that may transmit data frombottom hole assembly 130 to an information handling system 138 atsurface 108. Information handling system 138 may comprise one or more ofa personal computer 141, a video display 142, a keyboard 144 (i.e.,other input devices.), and non-transitory computer-readable media 146(e.g., optical disks, magnetic disks) that can store code representativeof the methods described herein. In addition to, or in place ofprocessing at surface 108, processing may occur downhole. As discussedbelow, methods may be utilized by information handling system 138 forstochastic trajectory projection of the bottom hole assembly 130 ofdrilling system 100.

FIG. 2 depicts an exemplary flow diagram illustrating a method fordetermining one or more stochastic trajectory projections and confidenceregions for a bottom hole assembly 130 based at least in part on one ormore of system model parameters, steering inputs, bottom hole assemblyinitial conditions, working mode selection and settings, and a desirednumber of trajectory projections. In one or more embodiments, thestochastic trajectory projection module 210 for the steering model ofdrilling system 100 may be described in Equation (1), with a sequence ofsteering inputs 222 for upcoming drilling footage.

{dot over (x)}=f(x,u,p)   (1)

where x represents the initial conditions of the bottom hole assembly224, which may include one or more of inclination, azimuth, build rate,walk rate, true vertical depth and similar values; u represents steeringinputs 222, which may include one or more of steering ratios and toolface angles and which may be provided as a sequence of inputs; and pdenotes the system model parameter probability distributions 220. In oneor more embodiments, the steering inputs 222 may be derived using adynamic control scheme such as model predictive control. In one or moreembodiments, the steering inputs 222 may be quantitative valuesspecified by the drilling personnel or the control system. In one ormore embodiments, the probability distributions of p can be developed inreal-time using one or more parameter data sets. In one or moreembodiments, a parameter data set may comprise any one or more of areal-time (or online) calibration method or data analytics,non-real-time (or offline) calibration method or data analytics, any oneor more models of varying degrees of fidelity, and the experience of oneor more persons skilled in the art of drilling or control systems. Inone or more embodiments, system model parameter set p may depicted byEquation (3). In one or more embodiments, system model parameter set pmay be directly obtained from an online identification method. In one ormore embodiments, an identification method or system identification mayrefer to one or more methods of using one or more of measurements andknown external influence to determine one or more system modelparameters. In one or more embodiments, a known external influence maycomprise one or more system inputs. In one or more embodiments, the term“online” may be used to denote a real-time method or system in which amodel controller for the bottom hole assembly is operatingsimultaneously with and controlling the bottom hole assembly. An onlinemethod or system enables identification of one or more new values forsystem model parameter set p as one or more of new measurements andinputs are obtained. In one or more embodiments, the new measurementsand inputs may enable improved controller performance by refining andupdating prior measurements and inputs during one or more drillingoperations. In one or more embodiments, the term “offline” may besynonymous with a method or system that is not operating in real-time.In one or more embodiments, the system model parameter probabilitydistributions 220 may be assumed to follow a normal distribution, as isassumed in Equations (3)-(5). In one or more embodiments, the systemmodel parameters probability distributions 220 may be any alternativetype of distribution.

In one or more embodiments, the system model parameters probabilitydistributions 220 may be one or more of the elements of a steeringmodel. For example and without limitation, the steering model may beused to estimate the position of a drill bit 122 (depicted in FIG. 1 )and attitude of the drill bit may be represented by the followingdepth-based second order differential equation:

τ{umlaut over (θ)}=−{dot over (θ)}+K _(act) u+K _(bias), initialconditions: θ₀, {dot over (θ)}₀   (2)

identified as T is a depth constant, K_(act) is the magnitude of thebottom hole assembly 130 turning capability, K_(bias) represents boththe inherent steering tendency of bottom hole assembly 130 as well asany external forces on bottom hole assembly 130, θ₀ is the initial angle(inclination or azimuth), and θ₀ is the initial curvature (build rate orwalk rate). The dot notation in this equation represents a derivativewith respect to distance, not time. Without limitation, θ is also usedto represent a vector of the system model parameter probabilitydistributions 220. In one or more embodiments, the system modelparameter probability distributions 220 may be generated using priorexperience, knowledge of the subsurface formation and the equipment ofthe drilling system, prior analyses, and the like.

In one or more embodiments, model parameter set p may be described by amultivariate normal probability distribution using one or more of themodel parameter set's mean and variances or covariances. Themultivariate normal distribution of an n-dimensional parameter vectorP=(P, P₂, . . . P_(n)) may be written as:

P˜N(μ, K_(pp))   (3)

where μ is an n-dimensional mean vector:

μ=E[P]=(E[P ₁ ], E[P ₂ ], . . . , E[P _(n)])   (4)

and where K_(pp) is an n×n covariance matrix:

K _(pp) =E[(P _(i)−μ_(i)) (P _(j)−μ_(j))]  (5)

such that 1≤i and j≤n. That is, in one or more embodiments, theprobability distribution for K_(act) may have one or more interactionswith K_(bias). In one or more embodiments, the multivariate distributionmay be any alternative type of distribution.

As depicted in FIG. 2 , the stochastic trajectory projection module 210may be run with the specified input parameters to calculate theprojected trajectories. Disclosed herein are two working modes 226. Inthe first working mode, a single model is used to project the trajectoryof the bottom hole assembly throughout the entire depth horizon. A firstset of model parameters may be selected from the system model parameterprobability distributions 220 input to the stochastic trajectoryprojection module 210. The first set of model parameters may then usedin combination with the steering inputs 222 throughout the entire depthhorizon to generate the stochastic trajectory projections for the bottomhole assembly. The first working mode may be referred to as a one-modelmode. In one or more embodiments, the one-model mode may be useful as itmay provide a more responsive solution because these calculations may beperformed more quickly. The one-model mode may also be beneficial forshorter prediction horizons.

In the second working mode, multiple different models are used atdifferent depths within the projected. For a first predetermined depthinterval (or length of the borehole), a first set of model parametersmay be selected from the system model parameter probabilitydistributions 220 input to the stochastic trajectory projection module210. The first set of model parameters may then used in combination withthe steering inputs 222 to generate the stochastic trajectoryprojections for the bottom hole assembly for the first predetermineddepth interval. After projecting the stochastic trajectory projectionsfor the bottom hole assembly for the first predetermined depth interval,a second set of model parameters is selected from the system modelparameter probability distributions 220 input to the stochastictrajectory projection module 210 and used to generate the stochastictrajectory projections for the bottom hole assembly for the secondpredetermined depth interval. This process is repeated until the finaltrajectory for the bottom hole assembly, which is the combinedstochastic trajectory projections, traverses the entire depth horizon.The second working mode may be referred to as a multi-model mode. In oneor more embodiments, the multi-model mode may reduce the size of theconfidence regions of the stochastic trajectory projections. In one ormore embodiments, the multi-model mode may provide more confidentpredictions, across the depth horizon than a one-model model because,for example, the multi-model mode supports updates to the modelparameter set across the depth horizon.

The selection of the mode and other settings, including, withoutlimitation, the depth interval for stochastic trajectory projections maybe dependent on one or more factors, including, without limitation, anyprior data analyses or knowledge of one or more of the subsurfaceformation and the equipment of the drilling system. In one or moreembodiments, the number of stochastic trajectory projections for theentire depth horizon may be provided as an input to the stochastictrajectory projection module 210.

In one or more embodiments, a number of stochastic trajectoryprojections over the entire depth horizon may be generated, where thenumber of stochastic trajectory projections 228 may be represented by Nand where N is a positive integer. In one or more embodiments, N may be100 or more. In one or more embodiments, a median or mean value for theN stochastic trajectory projections may be calculated. In one or moreembodiments, any outliers in the stochastic trajectory projections maybe identified and eliminated before the median or mean value iscalculated for the N stochastic trajectory projections.

In one or more embodiments, the N stochastic trajectory projections 228may be used to generate a vector including N data points, where eachdata point corresponds to particular position within the subsurfaceformation. In one or more embodiments, the position within thesubsurface formation may comprise a depth point, horizontal point, orany other identifier for the location of the position within thesubsurface formation. Quantiles between the cumulative probabilities ofq1 and q2 of the elements in the vector can be calculated where q1, q2are in the interval of [0, 1] and q1<q2. With quantiles derived at eachdepth point, a confidence region is established in which theabovementioned resulting trajectory is at the center. For example, a 95%confidence region means a pair of 2.5% and 97.5% quantiles need to becalculated and the remaining data points in the vector are used todetermine the confidence region. Multiple confidence regions orquantiles can be obtained and plotted and presented in the same orseparate displays.

In one or more embodiments, one or more of data, analysis, experience,and knowledge may be used to select identify desired confidence regions,where smaller confidence regions generally indicate greater confidencethat the actual trajectory of the bottom hole assembly will align withthe stochastic trajectory projections. In one or more embodiments, amulti-model mode may generate narrower confidence regions, indicatingincreased confidence in the multi-model trajectory projections than inthe stochastic trajectory projections generated by a one-model mode. Forexample and without limitation, a multi-model trajectory projection maygenerate narrower confidence regions by better representing thevariations in subsurface operating conditions based on generation ofmultiple trajectory projections using multiple models from the systemmodel parameter probability distributions 220. In one or moreembodiments, the number of models utilized may affect the area ofconfidence regions given the same distributions. For example and withoutlimitation, over a given depth horizon, a one-model mode may use only asingle model parameter set drawn from data set p while a multi-modelmode may use, for example, ten parameter sets drawn from model parameterset p for the same depth horizon. Accordingly, it may be desirable tosample multiple models from data set p to attempt to generate a betterrepresentation of the probability distribution of model parameters.

In summary, as shown in FIG. 1 , the stochastic trajectory projectionmodule 210 may receive inputs specifying the model parameter probabilitydistributions. In one or more embodiments, these inputs may be one ormore established system model parameter probability distributionsacquired directly from any identification method discussed herein. Oneor more sets of model parameters may be obtained by randomly samplingfrom the model parameters probability distributions 220. After selectingthe desired working mode and the desired number of trajectoryprojections, a projected trajectory may be calculated using the selectedset or sets of model parameters, steering inputs 222, and bottom holeassembly initial conditions 224. This calculation process may berepeated according to the desired number of trajectory projections. Asdescribed above, the working mode (one-model or multi-model) and itssettings may be specified to achieve a desired confidence of thestochastic trajectory projection. The outputs of the stochastictrajectory projection module 210 are the projected trajectory of thebottom hole assembly and the confidence region, which may be provided inreal-time to, for example and without limitation, a trajectorycontroller 242 and a display 240. In one or more embodiments, thetrajectory controller 242 may comprise one or more of a system and amethod for producing control outputs that may guide the bottom holeassembly along, or as closely as possible, the target well, which may beprovided in a well plan, based at least in part on the outputs of thestochastic trajectory projection module 210, including at least theprojected trajectory 230 and projected confidence region 232.

In one or more embodiments, the projected trajectory 230 and projectedconfidence region 232 may be used for a variety of purposes. In one ormore embodiments, the projected trajectory 230 and projected confidenceregion 232 may be used to update future steering decisions, resulting innew steering inputs 222 to the stochastic trajectory projection module210. In one or more embodiments, an actual trajectory that poorly alignswith the stochastic projected trajectory 230 or lies outside theprojected confidence region 232 may indicate changes in the drillingconditions and signal an operator to recalibrate and update the modelparameter set drawn from data set p, for example and without limitation,by running the system identification again using the new measurements.In one or more embodiments, the projected trajectory 230 and projectedconfidence region 232 may be used to mitigate any possible risks of theborehole 102 interfering with other boreholes in a subsurface formation106 or falling behind the target. For example and without limitation, anoperator may provide new inputs to trajectory controller 242 if there isa significant overlap between the projected trajectory 230 and an areaof possible collision.

In an additional example without limitation, an operator may provide newinputs to trajectory controller 242 such as applying additional steeringpower if the projected trajectory 230 shows a risk of deviating from awell plan for borehole 102 and, in one or more embodiments, theadditional steering power can be repeatedly provided to the trajectorycontroller 242 if the projected trajectory 230 continues to show theborehole 102 deviating from the well plan. For example, if the actualborehole has begun to deviate from the well plan, the projectedtrajectory 230 and projected confidence region 232 for a given set ofsteering inputs 222 enables an operator or controller to determine thatthe projected trajectory 230 will return to or overlap with the wellplan.

FIG. 3 depicts an exemplary well plan and stochastic trajectoryconfidence regions for a borehole across a depth horizon. FIG. 3 aillustrates an exemplary trajectory of a wellbore as segments 301, 303,305, 307, and 309 with well plan 330 across depth horizon 350, whileFIG. 3 b illustrates an exemplary trajectory of a similar wellbore assegments 321, 323, 325, 327, and 329 with substantially the same wellplan 330 across the same depth horizon 350. In FIG. 3 a , the confidenceregion 311 is generated by stochastic trajectory projection module 210using a one-model mode is shown overlapping segment 307. An operator,using this exemplary figure, would have confidence that the projectedtrajectory of the bottom hole assembly will overlap with the trajectoryof the borehole in the well plan. As illustrated in FIG. 3 a , aone-model mode would use the model parameter set drawn from data set pacross the entire depth horizon. That is, the model parameter set tostochastically project segment 301 would be the same set used tostochastically project segments 303, 305, 307, and 309. In FIG. 3 b ,the confidence region 331, which is slightly narrower than confidenceregion 311, is similarly shown overlapping segment 327 after theconfidence region 331 is generated by stochastic trajectory projectionmodule 210 using a multi-model mode. In contrast to FIG. 3 a , each ofthe segments in FIG. 3 b would use a model parameter set drawn from dataset p than any other segment. That is, a first model parameter set drawnfrom data set p would be used to stochastically project the trajectoryfor segment 321, a second model parameter set drawn from data set pwould be used to stochastically project the trajectory for segment 323,a third model parameter set drawn from data set p would be used tostochastically project the trajectory for segment 325, a fourth modelparameter set drawn from data set p would be used to stochasticallyproject the trajectory for segment 327, and a fifth model parameter setdrawn from data set p would be used to stochastically project thetrajectory for segment 329. While confidence region 331 is shown as anarrower region than confidence region 311, this is merely illustrativethe circumstances of drilling operations, subsurface conditions, andother factors may result in some one-model mode stochastic projectionsgenerating narrower confidence regions than comparable multi-model modestochastic projections.

FIGS. 4-5 depict exemplary stochastic trajectory projections for abottom hole assembly. The circles, 410, 420, 510, 520 represent controlpoints (or depth points) at which new inputs, which may comprise one ormore sets of model parameters, steering inputs 222, bottom hole assemblyinitial conditions 224, working mode selection and settings 226, and anumber of trajectory projections 228, are provided to the stochastictrajectory projection module 210 before a new projected trajectory 230and projected confidence region 232 are output to display 240 andtrajectory controller 242. See FIG. 2 . For example, as depicted inFIGS. 4-5 , a prediction depth horizon may be from a depth ofapproximately 6000 feet to a depth of approximately 6300 feet. Asdepicted in FIG. 4 , the left circle 410 specifies an initialinclination of the bottom hole assembly, while, in FIG. 5 , the leftcircle 510 specifies an initial azimuth of the bottom hole assembly.Using the process disclosed herein, stochastic trajectory projectionsfor inclination and azimuth are achieved (depicted using dashed lines)with 95% confidence regions (depicted as the shaded area). In FIGS. 4-4, an exemplary multi-model mode using 10 models for a stochastictrajectory projection. FIGS. 4-5 further depict a dashed-dot line 440between the circles 410, 420, and a dashed-dot line 540 between thecircles 510, 520 depicting actual trajectories for inclination andazimuth of the bottom hole assembly conforming closely to the stochastictrajectory projections 430, 530 and within the confidence regions 450,550 of the stochastic trajectory projection. As depicted in both FIGS.4-5 , the right circles 420, 520 are—about 90 feet deeper than the leftcircles 410, 510 and depict control points at which a new set of inputsmay be provided to the stochastic trajectory projection module 210 togenerate additional stochastic trajectory projections.

In one or more embodiments, a projected confidence region generated bythe stochastic trajectory projection module 210 using the one-model modemay be larger, and in some cases substantially larger, than theprojected confidence region generated by the stochastic trajectoryprojection module 210 using the multi-model mode.

In one or more embodiments, the stochastic trajectory projection module210 may use one or more stochastic simulation methods including, withoutlimitation, Monte Carlo simulation methods, to simulate and project thestochastic trajectory projections. In one or more embodiments, one ormore additional inputs may be provided to the stochastic trajectoryprojection module 210, including, without limitation, weight on bit,RPM, flow rate. These additional inputs may enable the stochastictrajectory projection module 210 to account for changes in one or moredrilling parameters and may thereby improve the quality of thestochastic trajectory projections.

FIG. 6 depicts an exemplary stochastic trajectory projection andconfidence regions for a bottom hole assembly across a depth horizon.Like FIGS. 4-5 , FIG. 6 a-6 b show the projected trajectories 610, 630and confidence regions 615, 620, 635, 640 generated by stochastictrajectory projection module 210. FIG. 6 a illustrates a stochastictrajectory projection 610 and two confidence regions 615, 620 for theinclination of the bottom hole assembly across a depth horizon fromapproximately 9775 feet deep to approximately 10075 feet deep. FIG. 6 billustrates a stochastic trajectory projection 630 and two confidenceregions 635, 640 for the azimuth of the bottom hole assembly across thesame depth horizon from approximately 9775 feet deep to approximately10075 feet deep. Both FIG. 6 a-6 b illustrate that using a multi-modelmode may result in increased confidence in the stochastic trajectoryprojection generated by the stochastic trajectory projection module 210,as shown by narrower confidence regions 615, 635 generated by using themulti-model mode in comparison with the wider confidence regions 620,640 generated by using the one-model mode.

FIG. 7 depicts a flow diagram for generating stochastic trajectoryprojections for a bottom hole assembly. In step 710, a set of systemmodel parameters 220 are provided to the stochastic trajectoryprojection module 210. In step 720, one or more steering inputs 222 arecollected and provided to the stochastic trajectory projection module210. In one or more embodiments, the one or more steering inputs 222 maybe collected from the trajectory controller 242. In one or moreembodiments, the one or more steering inputs 222 may be collected froman operator, for example, an individual that has experience withdrilling and production operations in or around the borehole environment100. In step 730, a stochastic trajectory projection is generated basedon a Monte Carlo method, for example, using the set of system modelparameters from model parameter set p and Equation (2). In step 740, thestochastic trajectory projection module 210 compares the total number ofstochastic trajectory projections generated against the specified numberof stochastic trajectory projections 228. If the total number ofstochastic trajectory projections generated by the stochastic trajectoryprojection module 210 does not meet or exceed the specified number ofstochastic trajectory projections 228, the method returns to step 730and an additional stochastic trajectory projection is generated based onthe Monte Carlo method for example, using Equation (2). Alternatively,if the total number of stochastic trajectory projections generated bythe stochastic trajectory projection module 210 does meet or exceed thespecified number of stochastic trajectory projections 228, the methodmoves to step 750 and a projected trajectory 230 and a projectedconfidence region 232 for the bottom hole assembly 130 is calculated. Instep 760, one or more of the projected trajectory 230 and a projectedconfidence region 232 may be provided to one or more of the display 240and the trajectory controller 242. In step 770, the projected trajectory230 and a projected confidence region 232 may be used to update futuresteering decisions. For example and without limitation, future steeringdecisions may be provided to step 720 such that the future steeringdecisions may be used for additional stochastic trajectory projectionsgenerated by the stochastic trajectory projection module 210.

FIG. 8 depicts a schematic diagram of example information handlingsystem 800, for example, for use with or in an associated boreholeenvironment, for example without limitation the borehole environment 100depicted in FIG. 1 . The information handling system 138 of FIG. 1 maytake a form similar to the information handling system 800. A processoror central processing unit (CPU) 801 of the information handling system800 is communicatively coupled to a memory controller hub (MCH) or northbridge 802. The processor 801 may include, for example a microprocessor,microcontroller, digital signal processor (DSP), application specificintegrated circuit (ASIC), or any other digital or analog circuitryconfigured to interpret and/or execute program instructions and/orprocess data. Processor 801 may be configured to interpret and/orexecute program instructions or other data retrieved and stored in anymemory such as memory 803 or hard drive 807. Program instructions orother data may constitute portions of a software or application, forexample, application 858 or data 854, for carrying out one or moremethods described herein. Memory 803 may include read-only memory (ROM),random access memory (RAM), solid state memory, or disk-based memory.Each memory module may include any system, device, or apparatusconfigured to retain program instructions and/or data for a period oftime (for example, non-transitory computer-readable media). For example,instructions from a software program or application 858 or data 854 maybe retrieved and stored in memory 803 for execution or use by processor801. In one or more embodiments, the memory 803 or the hard drive 807may include or comprise one or more non-transitory executableinstructions that, when executed by the processor 801, cause theprocessor 801 to perform or initiate one or more operations or steps.The information handling system 800 may be preprogrammed or it may beprogrammed (and reprogrammed) by loading a program from another source(for example, from a CD-ROM, from another computer device through a datanetwork, or in another manner).

The data 854 may include treatment data, geological data, fracture data,microseismic data, mud candidate data, borehole imager measured data,inversion-estimated imaging properties, or any other appropriate data.The one or more applications 858 may include one or more machinelearning models, applications for one or more of down-sampling measureddata, calculating misfits or to minimize cost functions, to performpetrochemical inversions, to solve for formation permittivity, to alignmeasured data based on depth, azimuth, resolution, or any othermeasurement, extrapolating permittivity, scaling coefficients to matchborehole imager measurements with dielectric tool measurements,calculate dispersion curves of permittivity, calibrating coefficients,or any other appropriate applications. In one or more embodiments, amemory of a computing device includes additional or different data,application, models, or other information. In one or more embodiments,the data 854 may include treatment data relating to fracture treatmentplans. For example, the treatment data may indicate a pumping schedule,parameters of a previous injection treatment, parameters of a futureinjection treatment, or one or more parameters of a proposed injectiontreatment. Such one or more parameters may include information on flowrates, flow volumes, slurry concentrations, fluid compositions,injection locations, injection times, or other parameters. The treatmentdata may include one or more treatment parameters that have beenoptimized or selected based on numerical simulations of fracturepropagation. In one or more embodiments, the data 854 may include one ormore signals received by one or more transducers 136 a-c of FIG. 1 .

The one or more applications 858 may comprise one or more softwareprograms or applications, one or more scripts, one or more functions,one or more executables, or one or more other modules that areinterpreted or executed by the processor 801. For example, the one ormore applications 858 may include a fracture design module, a reservoirsimulation tool, a hydraulic fracture simulation model, or any otherappropriate function block. The one or more applications 858 may includemachine-readable instructions for performing one or more of theoperations related to any one or more embodiments of the presentdisclosure. The one or more applications 858 may includemachine-readable instructions for generating a user interface or a plot,for example, depicting fracture geometry (for example, length, width,spacing, orientation, etc.), pressure plot, hydrocarbon productionperformance. The one or more applications 858 may obtain input data,such as treatment data, geological data, fracture data, measurementdata, or other types of input data, from the memory 803, from anotherlocal source, or from one or more remote sources (for example, via theone or more communication links 814). The one or more applications 858may generate output data and store the output data in the memory 803,hard drive 807, in another local medium, or in one or more remotedevices (for example, by sending the output data via the communicationlink 814).

Modifications, additions, or omissions may be made to FIG. 8 withoutdeparting from the scope of the present disclosure. For example, FIG. 8shows a particular configuration of components of information handlingsystem 800. However, any suitable configurations of components may beused. For example, components of information handling system 800 may beimplemented either as physical or logical components. Furthermore, insome embodiments, functionality associated with components ofinformation handling system 800 may be implemented in special purposecircuits or components. In other embodiments, functionality associatedwith components of information handling system 800 may be implemented inconfigurable general-purpose circuit or components. For example,components of information handling system 800 may be implemented byconfigured computer program instructions.

Memory controller hub 802 may include a memory controller for directinginformation to or from various system memory components within theinformation handling system 800, such as memory 803, storage element806, and hard drive 807. The memory controller hub 802 may be coupled tomemory 803 and a graphics processing unit (GPU) 804. Memory controllerhub 802 may also be coupled to an I/O controller hub (ICH) or southbridge 805. I/O controller hub 805 is coupled to storage elements of theinformation handling system 800, including a storage element 806, whichmay comprise a flash ROM that includes a basic input/output system(BIOS) of the computer system. I/O controller hub 805 is also coupled tothe hard drive 807 of the information handling system 800. I/Ocontroller hub 805 may also be coupled to an I/O chip or interface, forexample, a Super I/O chip 808, which is itself coupled to several of theI/O ports of the computer system, including a keyboard 809, a mouse 810,a monitor 812 and one or more communications link 814. Any one or moreinput/output devices receive and transmit data in analog or digital formover one or more communication links 814 such as a serial link, awireless link (for example, infrared, radio frequency, or others), aparallel link, or another type of link. The one or more communicationlinks 814 may comprise any type of communication channel, connector,data communication network, or other link. For example, the one or morecommunication links 814 may comprise a wireless or a wired network, aLocal Area Network (LAN), a Wide Area Network (WAN), a private network,a public network (such as the Internet), a WiFi network, a network thatincludes a satellite link, or another type of data communicationnetwork.

A memory or storage device primarily stores one or more softwareapplications or programs, which may also be described as program modulescontaining computer-executable instructions, which may be executed bythe computing unit for implementing one or more embodiments of thepresent disclosure. The memory, therefore, may include one or moreapplications including, for example, a transmitter control application,a receiver control application, and one or more applications enablingone or more of the processes or sub-processes illustrated in FIG. 2 andmay produce outputs like those shown in FIGS. 4-5 . These applicationsmay integrate functionality from additional or third-party applicationprograms or from system files stored in memory or on a storage device.An application may perform one or more of the steps in FIG. 2 . Systemfiles, such as an ASCII text file may be used to store the instructions,data input, or both for the applications as may be required in, forexample, one or more steps of FIG. 2 . In certain embodiments, any oneor more other applications may be used in combination. In certainembodiments, any one or more other applications may be used incombination may be used as stand-alone applications.

Although the computing device 800 is shown as having one or moregeneralized memories, the computing device 800 typically includes avariety of non-transitory computer readable media. By way of example,and not limitation, non-transitory computer readable media may comprisecomputer storage media and communication media. The memory may includecomputer storage media, such as a ROM and RAM in the form of volatilememory, nonvolatile memory, or both. A BIOS containing the basicroutines that help to transfer information between elements within thecomputing unit, such as during start-up, is typically stored in the ROM.RAM typically contains data, program modules, other executableinstructions, or any combination thereof that are immediately accessibleto, presently being operated on, or both by the processing unit. By wayof example, and not limitation, the computing device 800 may include anoperating system, application programs, other program modules, andprogram data.

The components shown in the memory may also be included in otherremovable/non-removable, volatile/nonvolatile non-transitory computerstorage media or the components may be implemented in the computingdevice 800 through an application program interface (“API”) or cloudcomputing, which may reside on a separate computing device coupledthrough a computer system or network (not shown). For example andwithout limitation, a hard disk drive may read from or write tonon-removable, nonvolatile magnetic media, a magnetic disk drive mayread from or write to a removable, nonvolatile magnetic disk, and anoptical disk drive may read from or write to a removable, nonvolatileoptical disk such as a CD-ROM or other optical media. Otherremovable/non-removable, volatile/nonvolatile computer storage mediathat may be used in the exemplary operating environment may include, butare not limited to, magnetic tape cassettes, flash memory cards, digitalversatile disks, digital video tape, solid state RAM, solid state ROM,or the like. The drives and their associated computer storage mediadiscussed above provide storage of computer readable instructions, datastructures, program modules, and other data for the computing unit.

The computing device 800 may receive commands or information from a userthrough one or more input devices such as the keyboard 809 and the mouse810. Additional input devices may comprise a microphone, joystick,touchscreen, scanner, voice or gesture recognition, one or more sensorsincluding one or more seismic sensors, and the like (not shown). Theseand other input devices may be coupled to the processing unit throughthe Super I/O chip 808 that is coupled to the ICH 805, but may becoupled by other interface and bus structures, such as a parallel portor a universal serial bus (USB) (not shown).

A monitor or other type of display device (not shown) may be coupled tothe MCH 802 via an interface, such as the GPU 804 or via Super I/O chip808. A graphical user interface (“GUI”) may also be used with the videointerface 804 to receive instructions from a user and transmitinstructions to the central processing unit 801. A GUI may be used todisplay the outputs of the processes described in in FIGS. 2 and 7 ,including, without limitation, as shown in FIGS. 4-5 , and may be usedto prompt or display modification of subsurface operations or productionactivities. The computing device 800 may comprise peripheral outputdevices such as speakers, printer, external memory, any other device, orany combination thereof, which may be coupled through any outputperipheral interface.

Any one or more input/output devices may receive and transmit data inanalog or digital form over one or more communication links 814 such asa serial link, a wireless link (for example, infrared, radio frequency,or others), a parallel link, or another type of link. The one or morecommunication links 814 may comprise any type of communication channel,connector, data communication network, or other link. For example, theone or more communication links 814 may comprise a wireless or a wirednetwork, a Local Area Network (LAN), a Wide Area Network (WAN), aprivate network, a public network (such as the Internet), a wirelessfidelity or WiFi network, a network that includes a satellite link, oranother type of data communication network.

Although many other internal components of the computing device 800 arenot shown, those of ordinary skill in the art will appreciate that suchcomponents and their interconnection are well known.

Any one or more embodiments of the present disclosure may be implementedthrough a computer-executable program of instructions, such as programmodules, generally referred to as software applications or applicationprograms executed by a computer. A software application may include, forexample, routines, programs, objects, components, data structures, anyother executable instructions, or any combination thereof, that performparticular tasks or implement particular abstract data types. Thesoftware application forms an interface to allow a computer to reactaccording to a source of input. For example, an interface applicationmay be used to implement any one or more embodiments of the presentdisclosure. The software application may also cooperate with otherapplications or code segments to initiate a variety of tasks based, atleast in part, on data received, a source of data, or any combinationthereof. Other applications or code segments may provide optimizationcomponents including, but not limited to, neural networks, earthmodeling, history-matching, optimization, visualization, datamanagement, and economics. The software application may be stored,carried, or both on any variety of memory such as CD-ROM, magnetic disk,optical disk, bubble memory, and semiconductor memory (for example,various types of RAM or ROM). Furthermore, the software application andone or more inputs or outputs may be transmitted over a variety ofcarrier media including, but not limited to wireless, wired, opticalfiber, metallic wire, telemetry, any one or more networks (such as theInternet), or any combination thereof.

Moreover, those skilled in the art will appreciate that one or more ofthe embodiments may comprise a variety of computer-systemconfigurations, including hand-held devices, multiprocessor systems,microprocessor-based or programmable consumer electronics,minicomputers, mainframe computers, and any combination thereof. Anynumber of computer-systems and computer networks are acceptable for usewith the present disclosure. The disclosure may be practiced indistributed-computing environments where tasks are performed byremote-processing devices that are linked through a communicationsnetwork. In a distributed-computing environment, program modules may belocated in both local and remote computer-storage media including memorystorage devices. The present disclosure may, therefore, be implementedin connection with various hardware, software, or any combinationthereof, in a computer system, information handling system, or otherprocessing system.

In one or more embodiments, a method for stochastically projecting awell trajectory of a bottom hole assembly in a subsurface formationcomprises receiving a first one or more system model parameters from asystem model parameter probability distribution, receiving a first oneor more steering inputs, receiving a first one or more valuescorresponding to the bottom hole assembly initial conditions at a firstposition within the subsurface formation, and stochastically projectinga first one or more trajectories of the bottom hole assembly from thefirst position within the subsurface formation to a second positionwithin the subsurface formation based at least in part on one or more ofthe first one or more system model parameters, the first one or moresteering inputs, and the first one or more values corresponding to thebottom hole assembly initial conditions.

In one or more embodiments, the method of further comprisesstochastically projecting the first confidence region between the firstposition and the second position based at least in part one or more ofthe first one or more stochastically projected trajectories, the firstone or more system model parameters, the received one or more steeringinputs, and the received one or more values corresponding to the bottomhole assembly initial conditions. In one or more embodiments, the methodfurther comprises providing one or more of the first one or morestochastically projected trajectories and the first confidence region toone or more of a display and a trajectory controller. In one or moreembodiments, the method further comprises discarding one or moreoutliers in the first one or more stochastically projected trajectoriesof the bottom hole assembly before stochastically projecting the firstconfidence region. In one or more embodiments, the method furthercomprises advancing the bottom hole assembly from the first position tothe second position. In one or more embodiments, the method furthercomprises stochastically projecting a second one or more trajectories ofthe bottom hole assembly from the second position to a third positionbased at least in part on one or more of the first one or more systemmodel parameters, the received one or more steering inputs, and thereceived one or more values corresponding to the bottom hole assemblyinitial conditions. In one or more embodiments, the method furthercomprises receiving a second one or more system model parameters fromthe system model parameter probability distribution, stochasticallyprojecting a second one or more trajectories of the bottom hole assemblyfrom the second position to a third position based at least in part onone or more of the second one or more system model parameters, thereceived one or more steering inputs, and the received one or morevalues corresponding to the bottom hole assembly initial conditions.

In one or more embodiments, the first one or more system modelparameters may be randomly selected from the system model parameterprobability distribution. In one or more embodiments, the method furthercomprises generating a second one or more one or more steering inputsand stochastically projecting a second one or more trajectories of thebottom hole assembly from the second position to a third position basedat least in part on one or more of the selected one or more system modelparameters, the second one or more steering inputs, and the received oneor more values corresponding to the bottom hole assembly initialconditions. In one or more embodiments, stochastically projecting thefirst one or more trajectories of the bottom hole assembly occurs inreal-time. In one or more embodiments, selecting a second one or moresteering inputs may be based at least in part on one or more of thefirst one or more stochastically projected trajectories and the firstconfidence region to one or more of a display and a trajectorycontroller. In one or more embodiments, the method further comprisesreceiving a second one or more system model parameters from the systemmodel parameter probability distribution, receiving a second one or moresteering inputs, receiving a second one or more values corresponding tothe bottom hole assembly initial conditions at a second position withinthe subsurface formation, and stochastically projecting a second one ormore trajectories of the bottom hole assembly from the second positionin the subsurface formation to a third position in the subsurfaceformation based at least in part on one or more of the second one ormore system model parameters, the second one or more steering inputs,and the second one or more values corresponding to the bottom holeassembly initial conditions.

In one or more embodiments, a system for stochastically projecting awell trajectory of a bottom hole assembly comprises a bottom holeassembly comprising one or more transducers, a trajectory controllercoupled to the bottom hole assembly, and an information handling systemcoupled to the transducers, where the information system comprises aprocessor, and a non-transitory computer readable medium for storing oneor more instructions that, when executed, causes the processor toreceive a first one or more system model parameters from a system modelparameter probability distribution, receive a first one or more steeringinputs, receive a first one or more values corresponding to the bottomhole assembly initial conditions from the one or more transducers at afirst position within a subsurface formation, and stochastically projecta first one or more trajectories of the bottom hole assembly from thefirst position within the subsurface formation to a second positionwithin the subsurface formation based at least in part on one or more ofthe first one or more system model parameters, the first one or moresteering inputs, and the first one or more values corresponding to thebottom hole assembly initial conditions.

In one or more embodiments, the one or more instructions, when executed,further causes the processor to stochastically project a confidenceregion for the projected trajectory of the bottom hole assembly betweenthe first position within the subsurface formation to the secondposition within the subsurface formation. In one or more embodiments,the system further comprises a display and the one or more instructions,when executed, further causes the processor to provide one or more ofthe first one or more stochastically projected trajectories and thefirst confidence region to one or more of the display and the trajectorycontroller. In one or more embodiments, the one or more instructions,when executed, further causes the processor to randomly select the firstone or more system model parameters from the system model parameterprobability distribution. In one or more embodiments, the one or moreinstructions, when executed, further causes the processor to one or moreof stochastically project the trajectory of the bottom hole assembly orstochastically project the confidence region for the projectedtrajectory of the bottom hole assembly in real time. In one or moreembodiments, the one or more instructions, when executed, further causesthe processor to receive a second one or more system model parametersfrom the system model parameter probability distribution, receive asecond one or more steering inputs; receive a second one or more valuescorresponding to the bottom hole assembly initial conditions at thesecond position within the subsurface formation, and stochasticallyproject a second one or more trajectories of the bottom hole assemblyfrom the second position in the subsurface formation to a third positionin the subsurface formation based at least in part on one or more of thesecond one or more system model parameters, the second one or moresteering inputs, and the second one or more values corresponding to thebottom hole assembly initial conditions.

In one or more embodiments, a method for stochastically projecting awell trajectory of a bottom hole assembly in a subsurface formation inreal time comprises receiving a first one or more system modelparameters from a system model parameter probability distribution,receiving a first one or more steering inputs, receiving a first one ormore values corresponding to the bottom hole assembly initial conditionsat a first position within the subsurface formation, stochasticallyprojecting a first one or more trajectories of the bottom hole assemblyfrom the first position within the subsurface formation to a secondposition within the subsurface formation, advancing the bottom holeassembly from the first position to the second position, receiving asecond one or more system model parameters from the system modelparameter probability distribution, receiving a second one or moresteering inputs, receiving a second one or more values corresponding tothe bottom hole assembly initial conditions at a second position withinthe subsurface formation, and stochastically projecting a second one ormore trajectories of the bottom hole assembly from the second positionwithin the subsurface formation to a third position within thesubsurface formation. In one or more embodiments, the method furthercomprises stochastically projecting the first one or more confidenceregions based on the stochastically projected first one or moretrajectories of the bottom hole assembly between the first positionwithin the subsurface formation and the second position within thesubsurface formation and further comprising stochastically projectingthe second one or more confidence regions based on the stochasticallyprojected second one or more trajectories of the bottom hole assemblybetween the second position within the subsurface formation and thethird position within the subsurface formation.

While the present disclosure has been described in connection withpresently preferred embodiments, it will be understood by those skilledin the art that it is not intended to limit the disclosure to thoseembodiments. It is therefore, contemplated that various alternativeembodiments and modifications may be made to the disclosed embodimentswithout departing from the spirit and scope of the disclosure defined bythe appended claims and equivalents thereof. In particular, with regardsto the methods disclosed, one or more steps may not be required in allembodiments of the methods and the steps disclosed in the methods may beperformed in a different order than was described. Furthermore, nolimitations are intended to the details of construction or design hereinshown, other than as described in the claims below. It is thereforeevident that the particular illustrative embodiments disclosed above maybe altered or modified and all such variations are considered within thescope and spirit of the present disclosure. In particular, every rangeof values (for example, “from about a to about b,” or, equivalently,“from approximately a to b,” or, equivalently, “from approximately a-b”)disclosed herein is to be understood as referring to the power set (theset of all subsets) of the respective range of values. The terms in theclaims have their plain, ordinary meaning unless otherwise explicitlyand clearly defined by the patentee.

What is claimed is:
 1. A method for stochastically projecting a welltrajectory of a bottom hole assembly in a subsurface formation, themethod comprising: receiving a first one or more system model parametersfrom a system model parameter probability distribution; receiving afirst one or more steering inputs; receiving a first one or more valuescorresponding to the bottom hole assembly initial conditions at a firstposition within the subsurface formation; and stochastically projectinga first one or more trajectories of the bottom hole assembly from thefirst position within the subsurface formation to a second positionwithin the subsurface formation based at least in part on one or more ofthe first one or more system model parameters, the first one or moresteering inputs, and the first one or more values corresponding to thebottom hole assembly initial conditions.
 2. The method of claim 1further comprising stochastically projecting the first confidence regionbetween the first position and the second position based at least inpart one or more of the first one or more stochastically projectedtrajectories, the first one or more system model parameters, thereceived one or more steering inputs, and the received one or morevalues corresponding to the bottom hole assembly initial conditions. 3.The method of claim 2 further comprising providing one or more of thefirst one or more stochastically projected trajectories and the firstconfidence region to one or more of a display and a trajectorycontroller.
 4. The method of claim 2 further comprising discarding oneor more outliers in the first one or more stochastically projectedtrajectories of the bottom hole assembly before stochasticallyprojecting the first confidence region.
 5. The method of claim 1 furthercomprising advancing the bottom hole assembly from the first position tothe second position.
 6. The method of claim 5 further comprisingstochastically projecting a second one or more trajectories of thebottom hole assembly from the second position to a third position basedat least in part on one or more of the first one or more system modelparameters, the received one or more steering inputs, and the receivedone or more values corresponding to the bottom hole assembly initialconditions.
 7. The method of claim 5 further comprising receiving asecond one or more system model parameters from the system modelparameter probability distribution, stochastically projecting a secondone or more trajectories of the bottom hole assembly from the secondposition to a third position based at least in part on one or more ofthe second one or more system model parameters, the received one or moresteering inputs, and the received one or more values corresponding tothe bottom hole assembly initial conditions.
 8. The method of claim 1,wherein the first one or more system model parameters are randomlyselected from the system model parameter probability distribution. 9.The method of claim 1, further comprising generating a second one ormore one or more steering inputs and stochastically projecting a secondone or more trajectories of the bottom hole assembly from the secondposition to a third position based at least in part on one or more ofthe selected one or more system model parameters, the second one or moresteering inputs, and the received one or more values corresponding tothe bottom hole assembly initial conditions.
 10. The method of claim 1,wherein the stochastically projecting a first one or more trajectoriesof the bottom hole assembly occurs in real-time.
 11. The method of claim3 selecting a second one or more steering inputs based at least in parton one or more of the first one or more stochastically projectedtrajectories and the first confidence region to one or more of a displayand a trajectory controller.
 12. The method of claim 5 furthercomprising: receiving a second one or more system model parameters fromthe system model parameter probability distribution; receiving a secondone or more steering inputs; receiving a second one or more valuescorresponding to the bottom hole assembly initial conditions at a secondposition within the subsurface formation; and stochastically projectinga second one or more trajectories of the bottom hole assembly from thesecond position in the subsurface formation to a third position in thesubsurface formation based at least in part on one or more of the secondone or more system model parameters, the second one or more steeringinputs, and the second one or more values corresponding to the bottomhole assembly initial conditions.
 13. A system for stochasticallyprojecting a well trajectory of a bottom hole assembly, the systemcomprising: a bottom hole assembly comprising one or more transducers; atrajectory controller coupled to the bottom hole assembly. aninformation handling system coupled to the transducers, the informationsystem comprising: a processor, and a non-transitory computer readablemedium for storing one or more instructions that, when executed, causesthe processor to: receive a first one or more system model parametersfrom a system model parameter probability distribution; receive a firstone or more steering inputs; receive a first one or more valuescorresponding to the bottom hole assembly initial conditions from theone or more transducers at a first position within a subsurfaceformation; and stochastically project a first one or more trajectoriesof the bottom hole assembly from the first position within thesubsurface formation to a second position within the subsurfaceformation based at least in part on one or more of the first one or moresystem model parameters, the first one or more steering inputs, and thefirst one or more values corresponding to the bottom hole assemblyinitial conditions.
 14. The system of claim 13, wherein the one or moreinstructions that, when executed, further causes the processor tostochastically project a confidence region for the projected trajectoryof the bottom hole assembly between the first position within thesubsurface formation to the second position within the subsurfaceformation.
 15. The system of claim 14, wherein the system furthercomprises a display and wherein the one or more instructions that, whenexecuted, further causes the processor to provide one or more of thefirst one or more stochastically projected trajectories and the firstconfidence region to one or more of the display and the trajectorycontroller.
 16. The system of claim 13, wherein the one or moreinstructions that, when executed, further causes the processor torandomly select the first one or more system model parameters from thesystem model parameter probability distribution.
 17. The system of claim14, wherein the one or more instructions that, when executed, furthercauses the processor to one or more of stochastically project thetrajectory of the bottom hole assembly or stochastically project theconfidence region for the projected trajectory of the bottom holeassembly in real time.
 18. The system of claim 13, wherein the one ormore instructions that, when executed, further causes the processor to:receive a second one or more system model parameters from the systemmodel parameter probability distribution; receive a second one or moresteering inputs; receive a second one or more values corresponding tothe bottom hole assembly initial conditions at the second positionwithin the subsurface formation; and stochastically project a second oneor more trajectories of the bottom hole assembly from the secondposition in the subsurface formation to a third position in thesubsurface formation based at least in part on one or more of the secondone or more system model parameters, the second one or more steeringinputs, and the second one or more values corresponding to the bottomhole assembly initial conditions.
 19. A method for stochasticallyprojecting a well trajectory of a bottom hole assembly in a subsurfaceformation in real time, the method comprising: receiving a first one ormore system model parameters from a system model parameter probabilitydistribution; receiving a first one or more steering inputs; receiving afirst one or more values corresponding to the bottom hole assemblyinitial conditions at a first position within the subsurface formation;stochastically projecting a first one or more trajectories of the bottomhole assembly from the first position within the subsurface formation toa second position within the subsurface formation; advancing the bottomhole assembly from the first position to the second position; receivinga second one or more system model parameters from the system modelparameter probability distribution; receiving a second one or moresteering inputs; receiving a second one or more values corresponding tothe bottom hole assembly initial conditions at a second position withinthe subsurface formation; and stochastically projecting a second one ormore trajectories of the bottom hole assembly from the second positionwithin the subsurface formation to a third position within thesubsurface formation.
 20. The method of claim 19, further comprisingstochastically projecting a first one or more confidence regions basedon the stochastically projected first one or more trajectories of thebottom hole assembly between the first position within the subsurfaceformation and the second position within the subsurface formation andfurther comprising stochastically projecting a second one or moreconfidence regions based on the stochastically projected second one ormore trajectories of the bottom hole assembly between the secondposition within the subsurface formation and the third position withinthe subsurface formation.